Not coring.....
Over my short span of handling cues I guess it's about 50-50 on BE cues that I've handled that have moved off center. Maaaaybe the guess a bit on the high side but I've concluded the majority, if any that had moved, had not been cored. Some I was indecisive on but based on the history of the builder I was guessing no.
Some worse than others but any off center moving is not a good program for long term stability. Just because you core doesn't ensure absolute non-movement but I like dealing with better odds and quality coring does that for me.
Aligning the end grain of the core with the face grain of the host is a good insurance program in mine and in my customers book.